We propose a new mechanism for prompt simultaneous production of $J/\psi$ and $\Upsilon$ mesons in high energy hadronic collisions. The process is considered as a perturbative production of $B_c^{(*)}$ mesons $g{+}g\to B_c^{(*)}{+}\bar{B_c}^{\!\!(*)}$ followed by a long-distance final state interaction that rearranges the quarks to form $J/\psi$ and $\Upsilon$ mesons. Passing from $B_c^{(*)}{+}\bar{B_c}^{\!\!(*)}$ configuration to $J/\psi{+}\Upsilon$ configuration may proceed via a hypothetical resonance state, the tetraquark. The goal of this work is to examine whether the respective cross section is large enough to encourage a direct detection of the tetraquark at the LHC conditions (yes), and whether this hypothesis can help to explain recent D0 data without assigning an unusually low value to $\sigma_{\rm eff}$ in the double parton scattering mechanism (no).